Praise be to Allaah.
According to the most correct opinion, it is not permissible to divorce a woman without a valid reason, because this is unjust to the woman and causes the blessing of marriage to be lost for no reason, and the break up of a family with which Allaah had blessed the sons of Adam, as Allaah says (interpretation of the meaning):
And among His Signs is this, that He created for you wives from among yourselves, that you may find repose in them, and He has put between you affection and mercy. Verily, in that are indeed signs for a people who reflect [al-Room 30:21]
Obedience to one's parents should only be with regard to things that are right and good, that Allaah and His Messenger love. It is not permissible to obey them with regard to something that Allaah and His Messenger have forbidden, as the Prophet (peace and blessings of Allaah be upon him) said: There is no obedience to anyone if it involves disobedience towards Allaah; obedience is only with regard to what is right and good. (Narrated by al-Bukhaari and Muslim from Ali, may Allaah be pleased with him).
And Allaah says (interpretation of the meaning);
But if they (both) strive with you to make you join in worship with Me others that of which you have no knowledge, then obey them not; but behave with them in the world kindly, and follow the path of him who turns to Me in repentance and in obedience [Luqmaan 31:15]
No kafaarah is required from either the mother or the father, apart from tawbah (repentance) and istighfaar (seeking forgiveness); they should also try to put matters right and bring people together again. If they do this, they will have a reward (from Allaah). Allaah says (interpretation of the meaning):
There is no good in most of their secret talks save (in) him who orders Sadaqah (charity in Allaahs Cause), or Maroof (Islâmic Monotheism and all the good and righteous deeds which Allaah has ordained), or conciliation between mankind; and he who does this, seeking the good Pleasure of Allaah, We shall give him a great reward [al-Nisa 4:114]
And Allaah knows best.